Pompeiu problem

Let be a Hausdorff topological space (cf. also Hausdorff space; Topological space), a non-negative Radon measure on , and a topological group of continuous self-mappings of leaving invariant. For and , denotes the action of on . A family of compact subsets of is said to have the Pompeiu property if the linear mapping given by

(a1)

is injective.

A typical example occurs when , is the Lebesgue measure, and is the Euclidean group of orientation-preserving rigid motions. Let denote the characteristic function of and its Fourier transform, which is an entire function of exponential type (cf. also Entire function) in . In this case one can prove [a7], [a12] that has the Pompeiu property if and only if

(a2)

When is the ball, (a2) can never be satisfied but if , a pair of balls of radii and (the centre plays no role in this case), then it has the Pompeiu property if and only if , where is the set of fractions for which and are positive roots of . (Here, is the Bessel function of the first kind and order , cf. Bessel functions.)

The key statement of the equivalence of the Pompeiu property with (a2) holds when is an irreducible symmetric space of rank and non-compact type and is its group of orientation-preserving isometries [a12].

When is a bounded open set with Lipschitz boundary in such that is connected, and if , then failure of the Pompeiu property for the singleton is equivalent to the existence of an eigenvalue for the overdetermined Neumann boundary value problem for the Euclidean Laplacian (cf. also Neumann boundary conditions)

(a3)

It was shown by S. Williams [a9] that if (a3) has a solution, then must be real-analytic, which allows for many positive examples of the Pompeiu property. The equivalence between (a3) and the failure of Pompeiu property and Williams' observation also holds when is a non-compact symmetric space of rank with the invariant Laplacian in this case [a4].

The natural conjecture that the existence of a solution for (a3) is equivalent to being a Euclidean ball is usually called Schiffer's conjecture. For instance, [a1] contains the result that for convex planar sets the existence of infinitely many eigenvalues for (a3) implies that is a disc. This inspired work of M. Agranovsky, C.A. Berenstein and P.C. Yang, N. Garofalo and F. Segala, T. Kobayashi, and others. See the excellent bibliographic survey [a11] for the details on the progress made on this conjecture up to date (1998), as well as general background on the Pompeiu property.

For with , condition (a2) is only known to be necessary, due to the failure of the spectral synthesis, [a7]. For instance, for , except for elementary examples of the type of three squares with sides parallel to the axes and sizes , , none of whose quotients is rational, one can show that if one takes to be, e.g., a rectangle, then (a2) is also sufficient for the Pompeiu property [a6].

This case of the Pompeiu problem has many applications in image and signal processing and leads to the problem of deconvolution, that is, given a finite family , find distributions of compact support such that

which amounts to finding a left inverse of the Pompeiu mapping (a1). See [a3], [a5], [a8] for details.

There is a local variant of the Pompeiu problem. For instance, let be the unit ball of , let be a pair of balls centred at the origin such that and , then the values of all the integrals

are enough to determine any function (the Berenstein–Gay theorem); see [a2], [a10], [a11], for additional references, extensions, as well as relations to the deconvolution problem mentioned earlier.

References

[a1]	C. Berenstein, "On the converse to Pompeiu's problem" Noteas e Communicaçoes de Mat. Univ. Fed. Pernambuco (Brazil) , 73 (1976)
[a2]	C. Berenstein, "The Pompeiu problem, What's new" R. Deville (ed.) et al. (ed.) , Complex Analysis, Harmonic Analysis and Applications , Res. Notes Math. , 347 , Pitman (1996) pp. 1–11
[a3]	C. Berenstein, E.V. Patrick, "Exact deconvolution for multiple operators" IEEE Proc. Multidimensional Signal Proc. , 78 (1990) pp. 723–734
[a4]	C. Berenstein, M. Shahshahani, "Harmonic analysis and the Pompeiu problem" Amer. J. Math. , 105 (1983) pp. 1217–1229
[a5]	C. Berenstein, D. Struppa, "Complex analysis and convolution equations" G.M. Henkin (ed.) , Encycl. Math. Sci. , 54 (1993) pp. 1–108
[a6]	C. Berenstein, B.A. Taylor, "The three-squares theorem for continuous functions" Arch. Rat. Mech. Anal. , 63 (1977) pp. 253–259
[a7]	L. Brown, B. Schreiber, B.A. Taylor, "Spectral synthesis and the Pompeiu problem" Ann. Inst. Fourier , 23 : 3 (1973) pp. 125–154
[a8]	S. Casey, D. Walnut, "Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms" SIAM Review , 36 (1994) pp. 537–577
[a9]	S. Williams, "A partial solution to the Pompeiu problem" Math. Ann. , 223 (1976) pp. 183–190
[a10]	L. Zalcman, "Offbeat integral geometry" Amer. Math. Monthly , 87 (1980) pp. 161–175
[a11]	L. Zalcman, "A bibliographic survey of the Pompeiu problem" B. Fuglede (ed.) et al. (ed.) , Approximation by solutions of partial differential equations , Kluwer Acad. Publ. (1992) pp. 185–194 (Addendum available from the author)
[a12]	C.A. Berenstein, L. Zalcman, "The Pompeiu problem in symmetric spaces" Comment. Math. Helvetici , 55 (1980) pp. 593–621